1.2 The Optical and Mechanical Systems
1.2.4 Modes and Symmetries of the Projected Mirror Portions
inversion symmetries known as the point group of the system. We will now turn to analyzing those symmetries in nanobeams and examining how they affect the modes of the system.
1.2.4 Modes and Symmetries of the Projected Mirror Por-
1.2.4.1 Photonic and Phononic Bands of the Projection
The projection of the nanobeam optomechanical crystal has discrete periodicity, sat- isfying n2(r) =n2(r+lΛˆx) and c(r) =c(r+lΛˆx), where Λ is the periodicity of the lattice and l is an integer.
Consider an operator that performs discrete translational coordinate transforma- tions on functons of any tensorial rank, the shift of length Λ being in the ˆxdirection,
TˆΛf(r)≡f(r+ Λˆx). (1.52) For the projection, ˆTΛn2(r) = n2(r) and ˆTΛc(r) = c(r) by definition. Together with the fact that differentials are not affected by constant shifts of the coordinate system, this also implies ˆTΛΞˆm = ˆΞm and ˆTΛΞˆm = ˆΞm in the projection. From this, it is easy to show that [ ˆT ,Ξˆm] = 0 and [ ˆT ,Ξˆm] = 0. From § 1.2.3.4, we know that the eigenvectors of the system are also eigenvectors of ˆTΛ, and we can classify the solutions according to their eigenvalues of ˆTΛ.
From Bloch’s theorem17, we know that the solutions of the can be expressed as
F(r) = u(r)eikx , (1.53)
where u(r) =u(r+mΛˆx) is called the Bloch function. This implies that
TˆΛF(r) =eikΛF(r). (1.54) These solutions can thus be classified according to a wave vector, kˆx. Because of the periodicity of eikΛ, all unique eigenvalues are contained in the domain, k ∈ [−π/Λ, π/Λ], which is called the first Brillouin zone18. The Bloch function will sat- isfy a separate differential equation that determines the frequency of the mode as a function of thek-vector. As the Bloch function is restricted to a finite region of space
17See, for instance, [14].
18Time-reversal symmetry and the fact that the frequencies (which are eigenvalues of Hermetian operators) are necessarily real guarantee that positive and negative wave vectors yield identical solutions. This allows the solutions to be further restricted to the first half of the first Brillouin zone,k∈[0, π/Λ].
(the unit cell) by its periodicity, the solutions for a givenk have a discrete spectrum of eigenfrequencies. Thus the eigenfrequencies for the acoustic and electromagnetic modes of the mirror form bands, which we will label by a band index19,n. A diagram that gives the spectrum of frequencies as a function of k in the first Brillouin zone20 will be called the band diagram of the structure.
1.2.4.2 Mirror Symmetries of the Projection
In addition to translational symmetry, the projection is also symmetric about the y= 0 andz = 0 planes, as shown in Fig. 1.6. As in§ 1.2.3.1 and§ 1.2.3.5, the mirror operators, σy and σz, which have covariant representations
σy =
1 0 0
0 −1 0
0 0 1
σz =
1 0 0
0 1 0
0 0 −1
, (1.55)
commute with the operators ˆΞm and ˆΞo. The electromagnetic and acoustic modes in the projection can thus be further classified with respect to their vector parity about these planes, each solution having an eigenvalue of the mirror operator such that σjQ(σ−1r) = pjQ(r), where j can be y or z, and pj = ±1. We accordingly classify the solutions to the wave equation by the wave vector k ∈[0, π/Λ], py, and pz.
The band diagram for the acoustic modes of a nanobeam’s projection is shown in Fig. 1.7, with the first ten band indices, n, labeled a to j, pz indicated by color, and py indicated by line shape. The mechanical displacement profiles of the unit cell are shown for each band at Γ and X. In the band diagram, the mirror symmetry σz, (across the plane defined by z = 0) is indicated by color: red corresponds to even vector parity (pz = 1) and blue to odd vector parity (pz = −1). Mirror symmetry σy (across the plane defined by y = 0) plane is indicated by the line shape: solid corresponds to even vector parity (py = 1) and dashed to odd vector parity (py =−1).
19Although the band indices may label the frequencies at a given value ofkin order of increasing frequency, the bands may cross; so this will not hold in general for all values ofk.
20The two high symmetry points of the first Brillouin zone in a 1D periodic structure are often assigned the names Γ fork= 0 andX fork=π/Λ.
Λ 0 Normalized |Q|1
Figure 1.7: Mechanical band diagram and corresponding normalized displacement profiles of the unit cell at the Γ (k = 0) andX (k=π/Λ) points. Color and linestyle indicate the symmetries with respect to ˆσz and ˆσy, respectively (see text for details).
The mechanical mode profiles are all viewed from a direction normal to the z = 0 plane unless labeled “yz”, in which case the viewing angle is normal to the x = 0 plane. The pinch, accordion, and breathing mode bands are b, i, and j, respectively.
As torsional modes can be difficult to interpret without isometric views, it is noted for the reader that the mechanical modes for band e at X, band f at Γ, and band h atX are all torsional mechanical modes.
The band diagram for the optical modes of a nanobeam’s projection is shown in Fig. 1.8, with the first four band indices, n, labeled a to d, and py indicated by line shape21. As will be the case in all of this work, the structures are much
21As in the case of the Fig. 1.7, solid corresponds to even vector parity (py = 1) and dashed to odd vector parity (py=−1).
0 0.5 1 0
50 100 150 200 250
Optical Frequency (THz)
kx/( π/Λ )
a b c
d a
Light Cone b
x y
Figure 1.8: Optical band diagram and corresponding normalized displacement profiles of the unit cell at the Γ (k = 0) andX (k =π/Λ) points. Color and linestyle indicate the symmetries with respect to ˆσz and ˆσy, respectively (see text for details). The field profiles shown correspond to Ey(x, y, z = 0).
thinner than they are wide, which makes the energy required to have odd z vector parity very large; thus the TM-like (z-odd) modes of the structure do not exist at relevant frequencies. The profile of Ey(x, y, z = 0) for four unit cells are shown for the fundamental (valence) y-even and y-odd bands at the X point22.
The optical band diagram also displays another important feature of the optical modes: index guiding. The shaded area of the band diagram, which is called the light cone, corresponds to the region ωo > ck. The line itself, ωo = ck, is called the light line. Above the light line, the modes of the nanobeam are propagating in the direction transverse to the waveguide (x). Below the light line, the modes are guided by the index contrast between the material and the air. The concept of index guiding will play a critical role in the localization of optical modes.
22The conduction band modes look very similar but have their maxima in the air and nodes in the material